Mission 4 Summary
Mission 4 Summary
Embedded Files
Learning Objectives
Identify and draw points, lines, line segments, rays, and angles. Recognize them in various contexts and familiar figures.
Identify, define, and draw perpendicular lines.
Identify, define, and draw parallel lines.
Use a circular protractor to understand a 1-degree angle as _1 360 of a turn. Explore benchmark angles using the protractor.
Measure and draw angles. Sketch given angle measures, and verify with a protractor.
Use the addition of adjacent angle measures to solve problems using a symbol for the unknown angle measure.
Recognize lines of symmetry for given two-dimensional figures. Identify line-symmetric figures, and draw lines of symmetry.
Define and construct triangles from given criteria. Explore symmetry in triangles.
Classify quadrilaterals based on parallel and perpendicular lines and the presence or absence of angles of a specified size.
Draw, identify, and label points, a line segment, and a line:
Here we will try to understand concepts like point, a line, a line segment, a ray and angles.
1. Point:
Definition: A point represents an exact location in space. It has no length, width, or thickness—just position.
How to Draw:
Draw a small dot on your paper.
Label it with a capital letter, such as "A”.
Example: Point A
2. Line Segment:
Definition: A line segment is part of a line that has two endpoints. It is the shortest path between two points.
How to Draw:
Draw two points on the paper.
Connect them with a straight line.
Label the endpoints with capital letters, such as "A" and "B”.
Example: The straight line between two points A and B is a line segment.
3. Line:
Definition: A line is a straight path that extends infinitely in both directions. It has no endpoints.
How to Draw:
Draw a straight line with arrows on both ends to indicate that it extends infinitely.
Label two points on the line with capital letters, such as "A" and "B”.
Name the line using any two points on it (e.g., Line AB) or use a lowercase letter (e.g., line "m").
Example: Point A and B with arrows on both ends to indicate infinite length
4. Ray:
Definition: A ray is a part of a line that starts at one point and extends infinitely in one direction.
How to Draw:
Draw a point (the starting point).
From that point, draw a straight line that continues infinitely in one direction (add an arrow to the line).
Label the starting point with a capital letter, such as "A."
Label a second point along the line, such as "B," to name the ray.
How to Label:
Use the starting point and another point on the ray to name it (e.g., Ray AB). The arrow always points to the right when naming (→AB).
Example:
5. Angle:
Definition: An angle is formed when two rays share the same starting point (also called the vertex).
How to Draw:
Draw two rays that meet at a common starting point.
Label the common starting point (vertex) with a capital letter, such as "A."
Label the points on each ray with different capital letters (e.g., "B" and "C").
How to Label:
Angles are labeled using three points. The vertex must be in the middle (e.g., ∠BAC or ∠CAB).
You can also use a single letter if only one angle is present at that vertex (e.g., ∠A).
Example: Two rays AB and BC meet at point B to form ∠ABC.
Parallel Lines:
Parallel lines are the lines that do not intersect or meet each other at any point in a plane.
Key characteristics:
They are always parallel and are at equidistant from each other.
Parallel lines are non-intersecting lines.
Notation: The symbol for showing parallel lines is '||'.
For example: line m is parallel to line n. Line m || line n
Real-Life Examples:
Perpendicular Lines:
Perpendicular lines are two lines that intersect at a right angle (90 degrees). When two lines are perpendicular to each other, the angle formed at the point of intersection is exactly 90°, creating a perfect "L" shape.
Key characteristics:
Right Angle: The defining feature of perpendicular lines is the right angle (90°) formed where they meet.
Notation: Perpendicular lines are often denoted using the symbol "⊥". For example, if line AB is perpendicular to line CD, it is written as AB ⊥ CD.
For example: If you have a horizontal line and a vertical line that intersect at right angle, the two lines are perpendicular.
Line l is perpendicular to Line m (line l ⊥ line m)
The lines above meet at a right angle, so they are perpendicular.
Real-Life Examples:
The corners of a square or rectangle are formed by perpendicular lines.
The cross of the letter "T" or the intersection of streets in a city grid are examples of perpendicular lines.
Intersecting lines:
Intersecting lines are two or more lines that cross or meet at a single point. This point where they meet is called the point of intersection.
Key characteristics:
Intersection Point: The lines share exactly one point, known as the point of intersection.
Angle Formation: When lines intersect, they form angles at the point where they cross. These angles can vary in size (they may or may not form a right angle).
Position: Intersecting lines can meet at any angle, but they do not have to be perpendicular.
Example:
Here the point of intersection is A.
Real life examples:
1. Intersecting roads:
2. Pair of scissors:
Summary:
Summary:
A point is a location and is usually named with a capital letter (e.g., A).
A line segment connects two points and has definite endpoints (e.g., Line Segment AB).
A line goes on infinitely in both directions and can be labeled with two points or a lowercase letter (e.g., Line AB or Line m).
A ray has a starting point and extends infinitely in one direction, labeled with two points (e.g., →AB).
An angle is formed by two rays that share a vertex, labeled with three points (e.g., ∠BAC or simply ∠A if it's clear).
Perpendicular lines are two lines that intersect at a right angle (90 degrees).
Intersecting lines are two or more lines that cross or meet at a single point called the point of intersection.
B. Angle Measurement
A. Directions for Constructing a Paper Protractor.
To make a paper protractor, start by cutting a radius into both a red and a white paper circle. Place the red circle on top of the white one, lining up the cuts with the edge of your desk. Pinch the white circle below the cut to keep it in place. To show an angle, turn the white circle's edge counterclockwise to match the angle you want to measure.
Example:
To measure a 60-degree angle with your paper protractor, follow these steps:
1. Cut a radius into both a red and a white paper circle.
2. Stack the red circle on top of the white circle, aligning the radii with the edge of your desk.
3. Pinch the corner of the white circle directly below the slit to keep the circles together.
4. To show a 60-degree angle, turn the white circle so the edge aligns with the 60-degree mark on the red circle
B. Reason about the number of turns necessary to make a full turn with different fractions of a full turn:
Students learn about angles and turns using a paper protractor. They observe that a full turn can be divided into smaller fractions, such as one-quarter (90 degrees) and one-eighth (45 degrees) of a full turn.
Example: Four quarter-turns complete a full circle, while eight eighth-turns do the same. Students see that eighth-turns are smaller than quarter-turns. They also discover that a full turn equals 360 degrees and that a protractor measures angles in degrees. The protractor is marked with increments from 0 to 360 degrees, helping students understand and measure different angles.
C. Measure and draw benchmark angles with the protractor:
Using a circular protractor, students learn that a quarter-turn, or right angle, measures 90°, a half-turn measures 180°, a three-quarter-turn measures 270°, and a full rotation measures 360°. By aligning the protractor with the base segment of an angle and its vertex, students can accurately measure these angles.
Example: find that a right angle is 90°, and that two 45° angles add up to 90°, demonstrating how different angles combine to form larger ones.
D. Use varied protractors to distinguish angle measure from length measurement.
Students can use their bodies or whiteboards to model different types of angles. They demonstrate a right angle (90°), an obtuse angle (greater than 90°), and an acute angle (less than 90°). They also practice approximating angles of 30° and 60° with their arms. Students learn that a 180° angle is called a straight angle and use their knowledge to identify perpendicular and parallel walls in the classroom.
Example: When asked to show a right angle, students might stretch one arm straight up towards the ceiling and the other arm straight out to the side, forming an angle of 90°. To demonstrate an angle of approximately 30°, students would bring their arms closer together, making a small angle. When asked about a straight angle, they would extend their arms in opposite directions to form a straight line, representing a 180° angle.
E. Explore the effect of angle size on arc length. Distinguish between angle and length measurement.
Exploring how the size of an angle affects the length of an arc on circles of different sizes. They discover that while the angle measure (e.g., 90°) is the same, the length of the arc depends on the circle's size. A larger circle will have a longer arc for the same angle compared to a smaller circle. This demonstrates that angle measurement is independent of arc length; both large and small circles can have arcs measuring the same angle, but the actual length of the arcs differs with the circle's size.
Example: Draw a 90° arc on two circles—one large (Circle A) and one small (Circle B)—they will find that the arc on Circle A is longer than the arc on Circle B, even though both arcs represent the same 90° angle. This is because the circumference of Circle A is larger, so each 90° arc covers a greater distance around Circle A compared to Circle B. This illustrates that while the angle measure is the same, the length of the arc varies with the size of the circle.
F. Use a 180° protractor to verify angle measure.
Using a 180° protractor to verify that angles with different arc lengths can have the same measure. Measuring angles ∠C and ∠D, discovering that despite differences in arc lengths due to different circle sizes, both angles can measure the same number of degrees (e.g., 60°). This illustrates that angle measure is consistent regardless of the circle’s size; angles with the same degree measure are equal in size, even if their arcs or surrounding circles differ.
Example: Measure two angles, ∠C and ∠D, using a 180° protractor. They find that both angles measure 60°, even though ∠C has a longer arc compared to ∠D. This shows that while the arcs (and therefore the circles) are different sizes, the degree measure of the angles is the same. If ∠C and ∠D were placed on top of each other, their sides would align perfectly, demonstrating that they are equal angles despite their different arc lengths
G. Use multiple protractors to measure the same angle
use various protractors—both 360° and 180° protractors, with different sizes and tick mark intervals—to measure the same angle. They discover that despite differences in protractor size and arc length, the angle measurement remains consistent. For instance, all protractors show that ∠E measures 130°, illustrating that angle measurement is independent of the protractor’s size or the length of the angle's sides.
Example: Measuring ∠E using three different protractors: a large 360° protractor, a smaller 180° protractor, and a mini protractor. Despite the differences in size and the length of the arcs marked on each protractor, all three show that ∠E measures 130°.
H. Measure angles less than 180° using a circular and 180° protractor.
Measuring angles using both circular and 180° protractors to verify consistency in angle measurement. They practice aligning protractors accurately to determine the measure of an angle, such as ∠CAB, which shows that both types of protractors give the same result.
Example: ∠DEF measures 120° on both the circular and 180° protractors.
I. Measure an angle greater than 180° by subtracting from 360°.
Measuring an angle greater than 180°, students use a circular protractor to find the angle directly, or a 180° protractor to measure the smaller angle formed by extending the angle's arc. By subtracting the smaller angle's measure from 360°, they determine the measure of the larger angle. For instance, if the smaller angle measures 130°, subtracting this from 360° gives 230°, which is the measure of the larger angle, ∠QRS.
Example: Measure the angle ∠QRS, which is greater than 180°, students use a circular protractor and find that it measures 230°. To use a 180° protractor, they first measure the smaller angle formed by extending the arc, which is 130°. Subtracting 130° from 360° (the total degrees in a circle) gives 230°, confirming the measurement of ∠QRS.
J. Measure an angle greater than 180° by adding on to 180°.
To measure an angle greater than 180°, extend one side of the angle to create a straight line, which measures 180°. Measure the smaller acute angle formed by the extension, and then add this measurement to 180° to determine the total angle.
Example: If the smaller angle is 50°, then the full angle is 180° + 50° = 230
C. Problem Solving with the Addition of Angle Measures
Derive the angle measures of an equilateral triangle.
Step 1: Place squares around a central point. (Model.) Fit them like puzzle pieces. Point to the central point.
Step 2: Trace ∠XYZ. Measure of ∠XYZ is 90°
Step 3: Similarly take 4 quarter turns around the central point.
Step 4: An addition sentence for the sum of all the right angles in degrees.
90° + 90° + 90° + 90° = 360°
Therefore, the sum of angles around a central point is 360°
Definition: Use benchmark angle measures to show that angle measures are additive.
Angles can be added together. If you have a big angle, you can split it into smaller angles, and when you add those smaller angles together, you get the measure of the big angle.
Example:
Folding Paper to Make Angles:
Step 1: Fold a paper in half, bottom to top, then left to right. You now have a straight line, which is a straight angle measuring 180°.
Step 2: Fold it again so the edges match. You’ve made a right angle, which is 90°.
Step 3: Fold the corner down to meet the bottom edge, splitting the 90° angle into two equal parts. Each part is 45°, and together they make 90°: 45°+45°=90°45° + 45° = 90°45°+45°=90°.
Step 4: Open the folds to see four equal angles of 45° each, showing that 45°+45°+45°+45°=180°45° + 45° + 45° + 45° = 180°45°+45°+45°+45°=180°.
Definition:
Demonstrating that the Whole Angle Measure is the Sum of Its Parts: The measure of a whole angle is the sum of its parts. When you break a large angle into smaller angles, those smaller angles add up to the measure of the original angle. If you know one part, you can find the other part by subtracting the known angle from the whole angle.
Example:
Folding Paper to Show Angle Additivity:
Step 1: Fold a piece of paper to create a 90° angle.
Step 2: Fold a corner down so it doesn’t touch the bottom, creating two smaller angles. Open it up; now there are four angles.
Step 3: If one of the smaller angles is 27° and another is 63°, these angles add up to make a straight angle of 180°. This shows that all parts add up to the whole: 27°+63°+27°+63°=180°27° + 63° + 27° + 63° = 180°27°+63°+27°+63°=180°.
Step 4: Use a protractor to measure angles and find unknown parts. For example, if you have a 90° angle and one part is 60°, you can find the other part by subtracting: 90°−60°=30°90° - 60° = 30°90°−60°=30°.
Equation Example: If a straight angle (180°) has a known part of 132°, the unknown angle xxx can be found with the equation: 132°+x=180°132° + x = 180°132°+x=180°. Solving this, x=180°−132°=48°x = 180° - 132° = 48°x=180°−132°=48°.
Decomposing a 360° Angle: Understanding Angle Additivity
A 360° angle, also known as a full circle, can be broken down into smaller angles, and when you add up these smaller angles, they will always equal 360°. This shows that no matter how you divide the circle, all the parts will add up to the whole.
Example:
1. Breaking Down a 360° Angle:
Step 1: Imagine using pattern blocks to create a full circle (360°). If you remove one block, you create a missing angle. To find this missing angle, you can write an equation. For example, if the angles you know are 120° and 120°, your equation would be: 120°+120°+x=360°120° + 120° + x = 360°120°+120°+x=360°. To find xxx, add the known angles and subtract from 360°: 240°+x=360°240° + x = 360°240°+x=360°, so x=120°x = 120°x=120°.
Step 2: Another way to decompose a 360° angle is by drawing two intersecting lines, creating four smaller angles around a point. If two angles are 33° and 147°, you can find the total by adding: 33°+147°+33°+147°=360°33° + 147° + 33° + 147° = 360°33°+147°+33°+147°=360°.
Definition: Determining Unknown Angles in Intersecting Lines
When two lines intersect, they form angles that add up to 180° along a straight line. Angles directly across from each other are equal, and the sum of all angles around a point is 360°. You can find unknown angles by using known angles and simple subtraction or addition.
Example:
Finding Unknown Angles with Intersecting Lines:
Step 1: Draw two intersecting lines, one red and one blue. Label one angle as 20°.
Step 2: To find the unknown angle on the red line (∠x), use the equation: 180°−20°=160°180° - 20° = 160°180°−20°=160°. So, ∠x = 160°.
Step 3: Look at the blue line. To find the unknown angle (∠y) on the blue line, use what you know: 180°−160°=20°180° - 160° = 20°180°−160°=20°, so ∠y = 20°.
Step 4: To find another angle (∠z) on the red line, use the same logic: 180°−20°=160°180° - 20° = 160°180°−20°=160°, so ∠z = 160°.
Step 5: Draw another set of intersecting lines with one angle measuring 110°. The angles across from each other are equal, so the unknown angles are 70°, 110°, and 70°. This shows that opposite angles are always the same.
D. Two-Dimensional Figures and Symmetry
Definition of Symmetry: Symmetry refers to a shape being identical on both sides when divided along a line (called the line of symmetry).
Symmetry: Look at the picture of the butterfly given below
We can observe that the left side of the butterfly is similar to the right side.
Folded Symmetry:
The square had more folds than the rectangle. We folded the square four different ways, and the sides matched perfectly each time. The rectangle only matched when folded two ways. The rectangle folded into smaller rectangles, but the square folded into smaller rectangles and triangles!
Line of Symmetry: When we fold images along the line, both halves match exactly. This line is called a line of symmetry.
Example: Line of symmetry with the Alphabets
The alphabet H has two lines of symmetry.
The alphabet A has only one line of symmetry.
Lines of Symmetry in Two-Dimensional Figures
A line of symmetry is a line that divides a shape into two equal parts that are mirror images of each other. If you fold the shape along this line, both halves will match exactly.
What is a Line-Symmetric Figure?
A line-symmetric figure is a shape that has at least one line of symmetry. Examples include:
A square (4 lines of symmetry)
An equilateral triangle (3 lines of symmetry)
A circle (infinite lines of symmetry)
A rectangle (2 lines of symmetry, if all sides are equal)
Steps to Recognize Lines of Symmetry:
1. Look for lines that divide the shape into two identical parts.
2. Check if one side is the mirror image of the other.
3. Mark the line of symmetry using a straight line.
How to Draw Lines of Symmetry:
1. Identify the middle point or center of the shape.
2. Use a ruler to draw a straight line across the shape, ensuring both sides are equal.
3. Check by folding or imagining folding the shape to see if both sides align perfectly.
Examples:
Symmetrical Shapes:
1. Square
Lines of Symmetry: 4 (2 diagonals, 1 vertical, 1 horizontal)
Real-life examples of square symmetry can be found in various objects and designs around us. Below are some common examples:
1. Chessboard or Checkered Tiles
The symmetrical design of a chessboard or checkered tile floor has vertical, horizontal, and diagonal symmetry.
2. Windows with Crossbars
Many windows have a square shape divided into smaller squares or panes, showing symmetrical patterns.
3. Picture Frames
Square picture frames exhibit symmetry if their design is uniform on all sides.
4. Clock Faces
A square-shaped clock face often has symmetrical placements for numbers and decorations.
2. Circle
Lines of Symmetry: Infinite (any line passing through the center)
Real-life examples of circle symmetry can be found in various objects and designs around us. Below are some common examples:
1. Wheels and Tires
Wheels of cars, bicycles, and other vehicles are circular with symmetry in their spokes or design.
2. Plates and Bowls
Circular plates and bowls maintain symmetry around their center.
3. Coins
Coins are circular and exhibit symmetry in their design, regardless of rotation.
Rhombus: There are two folds in the rhombus.
Trapezoid: There is only one fold that matches in this trapezoid. A rectangle is a trapezoid, and it has two folds. A square is a trapezoid too, and it has four folds.
Triangles:
Discover the attributes of various triangles.
A triangle has three sides and three angles.
Triangles by side length:
Quadrilateral:
Characteristics of Quadrilateral
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